For this problem, my approach was to prove that $\sqrt{n}\mod 1$ is dense in $(0,1)$ and then extend the results to say $\sqrt{n}\mod\pi$ is dense in $(0,\pi)$. Once that is proved then the result follows immediately. However I'm not sure how to prove the first part.
By $\sqrt{n} \mod 1$, I mean the fractional part of $\sqrt{n}$ or $\sqrt{n} - \lfloor{\sqrt{n}}\rfloor$
By $\sqrt{n} \mod \pi$, I mean $\sqrt{n} - \pi\lfloor\frac{\sqrt{n}}{\pi}\rfloor$
